SCR denitration system unbiased model prediction control method based on dynamic variable gain Kalman filtering

Abstract

The invention discloses an unbiased model predictive control method for an SCR (Selective Catalytic Reduction) denitration system based on dynamic variable gain Kalman filtering. The unbiased model predictive control method comprises the following steps: establishing an augmented state space model of an SCR system; establishing a k-step forward prediction model; constructing a performance index optimization function; designing a Kalman filter for dynamically adjusting the gain according to the information; it is proved that model prediction control based on dynamic variable gain Kalman filtering can realize unbiased tracking of a set value in a steady state; for an SCR denitration system, a transfer function is used for describing the relation between the ammonia spraying amount and the NOx concentration of an outlet, and a control variable value obtained through calculation of the control method is output into a transfer function model. According to the method, the speed and accuracy of disturbance estimation are improved, the controller can act in time after disturbance changes on the basis of realizing unbiased tracking of the set value, the negative influence of the disturbance is suppressed, and the disturbance suppression effect is improved.

Description

Technical Field

[0001] The present invention belongs to the field of thermal automatic control, and particularly relates to an unbiased model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering. Background Art

[0002] During the operation of coal-fired power units, flue gas containing nitrogen oxides (NOx) is generated, which pollutes the environment. To ensure that the NOx content in the flue gas meets environmental protection requirements, selective catalytic reduction (SCR) technology has been widely applied in coal-fired power units. However, the SCR system has characteristics such as large inertia and multiple disturbances, and conventional proportional-integral-derivative (PID) control is difficult to achieve satisfactory control effects.

[0003] As a control algorithm based on rolling horizon optimization, model predictive control (MPC) has been widely applied in the control of SCR systems. Currently, various model predictive control strategies have been proposed for SCR systems, including DMC-PID cascade predictive control with multi-mode switching, MPC combined with neural networks to construct a predictive model, and MPC based on state space. However, due to the existence of a large number of unmeasurable disturbances in the SCR system, how to improve the ability of MPC to suppress unmeasurable disturbances still needs further research. Summary of the Invention

[0004] Object of the Invention: In order to overcome the deficiencies in the prior art, the present invention provides an unbiased model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering, which improves the speed and accuracy of disturbance estimation. On the basis of achieving unbiased tracking of the set value, the controller can act in a timely manner after the disturbance changes, suppress the negative impact of the disturbance, and improve the effect of disturbance suppression.

[0005] Technical Solution: To achieve the above object, the present invention provides an unbiased model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering, including the following steps:

[0006] S1: Establish an augmented state space model of the SCR system;

[0007] S2: Based on the augmented state space model of the SCR system, establish a k-step forward prediction model;

[0008] S3: Based on the k-step forward prediction model, construct a performance index optimization function;

[0009] S4: Design a Kalman filter that dynamically adjusts the gain according to the innovation;

[0010] S5: Apply the designed Kalman filter to model predictive control to achieve unbiased tracking of the set value at steady state through model predictive control based on dynamic variable-gain Kalman filtering;

[0011] S6: For the SCR denitration system, use the transfer function to describe the relationship between the ammonia injection amount and the outlet NOx concentration, set the controller parameters, perform optimization on the performance index optimization function, and output the calculated control variable value to the transfer function model.

[0012] Further, the expression of the augmented state space model of the SCR system in step S1 is as follows:

[0013]

[0014] where, x k is the state variable of the system; u k is the input variable, i.e., the ammonia injection amount; y k is the output variable, i.e., the NOx concentration at the outlet of the SCR system; k represents the current moment; A, B, C are system-related matrices, and Ο is the zero matrix;

[0015] Define the following variables:

[0016]

[0017] Then expression (1) is rewritten as:

[0018]

[0019] Further, the establishment process of the k-step forward prediction model in step S2 includes:

[0020] Define the increment of the input variable: Δu k = u k - u k-1 ; Considering the control time domain M and the prediction time domain P, and combining with formula (3), recursively obtain:

[0021]

[0022] where

[0023]

[0024] In the formula: is the predicted value of the output variable within the prediction time domain, ΔU k is the increment of the input variable within the control time domain; M is the control time domain, P is the prediction time domain, and M ≤ P.

[0025] Further, the performance index optimization function constructed in step S3 is as shown in formula (7):

[0026]

[0027] Among them, the performance indicators include the deviation between the system output and the set value and the increment of the input variable; in the formula is the output of the prediction model, Q and R are weight coefficient matrices, and Y r is the set value of the output of the controlled object within the prediction time domain; ΔU max , ΔU min are the upper and lower limits of the increment of the input variable within the control time domain respectively; U max , U min are the upper and lower limits of the input variable within the control time domain respectively; I is the identity matrix.

[0028] Furthermore, the Kalman filter that dynamically adjusts the gain according to the innovation in step S4 is shown in formula (8):

[0029]

[0030] Among them, is the estimated value of the augmented state variable, is the estimated value of the disturbance, NI is the innovation of the Kalman filter; K kal,x is the gain matrix of the filter estimated state, and K kal,d is the gain matrix of the filter estimated disturbance.

[0031] The filter gain matrix K kal is calculated as shown in formula (9):

[0032]

[0033] Among them, P is the covariance matrix, Q kal and R kal are covariance matrices, which respectively reflect the covariance situations of the process noise and the measurement noise; the definition of the fal function is shown in formula (10):

[0034]

[0035] Among them, α and δ are the parameters of the function, and e is the deviation; δ defines the size of the nonlinear interval, and α is used to adjust the nonlinear degree. Generally, 0 < α < 1 and δ > 0 are taken to achieve the effect of "small deviation, large gain; large deviation, small gain".

[0036] Furthermore, the conditions for the model predictive control based on the dynamic variable gain Kalman filter to achieve unbiased tracking of the set value under steady state in step S5 are as follows:

[0037] According to the existing theorem, if the following conditions are met, the linear MPC can achieve unbiased tracking of the set value:

[0038] Condition 1: The set value asymptotically converges to a constant value, the closed-loop system is asymptotically stable and converges to a constant value;

[0039] Condition 2: The augmented system described by Equation (1) is stable;

[0040] Condition 3: The number of disturbance variables is not greater than the number of system outputs;

[0041] Condition 4: The augmented model is observable;

[0042] Condition 5: The observer is asymptotically stable;

[0043] Condition 6: No constraints are active at steady state;

[0044] Condition 7: The optimization problem (7) has a feasible solution at all times.

[0045] Furthermore, the analysis of the unbiased model predictive control for the SCR denitration system based on the dynamic variable gain Kalman filter is carried out in step S5:

[0046] Proposition 1: If the Kalman filter is asymptotically stable, the Kalman filter gain of the disturbance estimation is full row rank, i.e., rank(K kal,d ) = n d , n d is the number of disturbance variables;

[0047] Proof: Consider the cases of |NI| > δ and |NI| ≤ δ separately

[0048] Case 1: |NI| ≤ δ, fal(NI, α, δ) = NI / δ 1-α

[0049] Denote K fal = 1 / δ 1-α , It follows that:

[0050]

[0051] Substitute Equation (9) into Equation (11), and it follows that:

[0052]

[0053] Since the filter is asymptotically stable and (1, 0) is not a pole, it follows that:

[0054]

[0055] Thus, it follows that is full row rank, K kal,d is full row rank, i.e., rank(K kal,d ) = n d ;

[0056] Case 2: |NI| > δ

[0057] Due to symmetry, consider the case where NI > δ. At this time, fal(NI, α, δ) = NI α , perform a Taylor expansion of fal(NI, α, δ) at NI = α:

[0058] fal(NI, α, δ) = NI α ≈ δ α + αδ α-1 (NI - δ) = αδ α-1 NI+(1 - α)δ α #(14)

[0059] Denote K fal2 = αδ α-1 , m = (1 - α)δ α , and obtain:

[0060]

[0061] Compare equation (15) with equation (11), and obtain:

[0062]

[0063] Similarly, since the filter is asymptotically stable and (1,0) is not a pole, then rank(K kal,d ) = n d ;

[0064] Proposition 2: Select n d = n y , n y is the number of output variables. If the closed-loop system and the filter are asymptotically stable, then the following equation holds:

[0065]

[0066] where are the estimated values of the state variables, input variables, output variables, and disturbances at steady state, respectively;

[0067] Proof: Since the closed-loop system and the filter are asymptotically stable, combining with equation (8) gives:

[0068]

[0069] Since n d = n y , K kal,d is a square matrix; at the same time, combining with Proposition 1, we obtain:

[0070]

[0071] Substitute equation (19) into equation (8), and obtain:

[0072]

[0073] By arranging equations (19) and (20), Proposition 2 is proven.

[0074] The proofs of the above two propositions ensure that fKF-MPC can achieve unbiased tracking of the setpoint.

[0075] Further, in step S6, setting the controller parameters includes the sampling time T s , the prediction horizon P, the control horizon M, the weight coefficient matrices Q, R, and the upper and lower limits U of the control variable and its increment max , U min , ΔU max , ΔU min .

[0076] Further, on the premise of satisfying the constraint conditions in formula (7), the controller solves the optimization problem at each sampling moment, and feeds the calculated control variable u into the SCR system represented by the transfer function model.

[0077] Based on the unbiased model predictive control design framework, the present invention applies the idea of dynamically adjusting the observer gain to the state space model predictive control, proposes a variable gain Kalman filter with a fal function, and proves that the model predictive control based on the dynamic variable gain Kalman filter can achieve unbiased tracking of the setpoint.

[0078] The present invention designs an unbiased model predictive control method for the SCR denitration system based on dynamic variable gain Kalman filtering, which improves the conventional Kalman filter by combining the fal function, so that the gain of the filter can be dynamically adjusted according to the innovation, and the accuracy of disturbance estimation is improved when the input end of the large inertia system such as SCR is disturbed.

[0079] The present invention can suppress the adverse effects of disturbances in the SCR system on the system output; when the innovation of the system is small during the occurrence of disturbances, the gain of the filter is dynamically amplified by combining the fal function, which can reduce the lag of the disturbance estimation process, so that the control action can be taken in time just after the disturbance starts to occur, reduce the dynamic deviation between the system output and the setpoint, and enhance the disturbance suppression ability.

[0080] Advantages: Compared with the prior art, the present invention proposes a method for dynamically adjusting the Kalman filter gain according to the innovation based on the fal function. For large-inertia systems such as SCR, it can estimate disturbances more quickly and accurately, making control actions more timely. Especially in the face of fast and randomly changing disturbances, when the disturbance first appears, since the gain of the filter is dynamically amplified when the innovation is small, the filter can estimate the disturbance faster and more accurately, which enables the control action to be taken in a timely manner after the disturbance begins to occur, significantly reducing the dynamic deviation between the system output and the reference, and effectively improving the ability to suppress disturbances. The method of the present invention has good anti-interference ability and robustness, can resist model mismatch and various interference factors encountered in the control process of the SCR system, suppress the adverse effects of internal disturbances of the system on system output parameters, and reduce the lag problem when the conventional Kalman filter estimates disturbances. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is the process flow chart of the SCR denitration system in the present invention;

[0082] Figure 2 is the block diagram of the unbiased model predictive control system of the SCR denitration system based on dynamic variable gain Kalman filtering in the present invention;

[0083] Figure 3 is the schematic diagram of the output variable and input variable curves of the unbiased model predictive control (fKF-MPC) based on dynamic variable gain Kalman filtering, PID, the model predictive control algorithm (MPC) based on state space, and MPC combined with disturbance observer (DOB-MPC) in this embodiment;

[0084] Figure 4 is the schematic diagram of the estimation effects of dynamic variable gain Kalman filtering, general Kalman filtering, and disturbance observer on disturbances in this embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0085] The following further clarifies the present invention in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. After reading the present invention, various equivalent modifications made by those skilled in the art fall within the scope defined by the appended claims of this application.

[0086] For the SCR denitration system process as Figure 1 shown, the present invention provides an unbiased model predictive control method for the SCR denitration system based on dynamic variable gain Kalman filtering. Referring to Figure 2 , it includes the following steps:

[0087] S1: Establish an augmented state space model of the SCR system, and the expression is as follows:

[0088]

[0089] Among them, x k is the state variable of the system; u k is the input variable, i.e., the ammonia injection amount; y k is the output variable, i.e., the NOx concentration at the outlet of the SCR system; k represents the current moment; A, B, and C are system-related matrices, and Ο is the zero matrix;

[0090] Define the following variables:

[0091]

[0092] Then expression (1) can be rewritten as:

[0093]

[0094] S2: Establish a k-step forward prediction model:

[0095] Define the increment of the input variable: Δu k = u k - u k-1 ; Considering the control horizon M and the prediction horizon P, and combining with equation (3), we can recursively obtain:

[0096]

[0097] Among them

[0098]

[0099] In the formula: is the predicted value of the output variable within the prediction horizon, ΔU k is the increment of the input variable within the control horizon; M is the control horizon, P is the prediction horizon, and M ≤ P.

[0100] S3: Construct a performance index optimization function, and the expression is as follows:

[0101]

[0102] Among them, the performance index includes the deviation between the system output and the set value and the increment of the input variable; in the formula is the output of the prediction model, Q and R are weight coefficient matrices, Y r is the set value of the output of the controlled object within the prediction horizon; ΔU max , ΔU min are the upper and lower limits of the increment of the input variable respectively; U max , U min are the upper and lower limits of the input variable within the control horizon respectively; I is the identity matrix.

[0103] S4: Design a Kalman filter that dynamically adjusts the gain according to the innovation:

[0104]

[0105] Among them, is the estimated value of the augmented state variable, is the estimated value of the disturbance, and NI is the innovation of the Kalman filter. K kal,x is the gain matrix of the filter's estimated state, and K kal,d is the gain matrix of the filter's estimated disturbance;

[0106] The filter gain matrix K kal is calculated as shown in formula (9):

[0107]

[0108] Among them, P is the covariance matrix, and Q kal and R kal are covariance matrices, respectively reflecting the covariance situations of the process noise and the measurement noise.

[0109] The definition of the fal function is shown in formula (10):

[0110]

[0111] Among them, α and δ are the parameters of the function, and e is the deviation. δ defines the size of the nonlinear interval, and α is used to adjust the nonlinear degree. Generally, 0 < α < 1 and δ > 0 are taken to achieve the effect of "small deviation, large gain; large deviation, small gain".

[0112] S5: Apply the designed Kalman filter to model predictive control and prove that the model predictive control based on dynamic variable-gain Kalman filtering can achieve unbiased tracking of the setpoint under steady state:

[0113] According to the existing theorem, if the following conditions are met, the linear MPC can achieve unbiased tracking of the setpoint:

[0114] Condition 1: The setpoint asymptotically converges to a constant value, the closed-loop system is asymptotically stable and converges to a constant value;

[0115] Condition 2: The augmented system described by formula (1) can be stabilized;

[0116] Condition 3: The number of disturbance variables is not greater than the number of system outputs;

[0117] Condition 4: The augmented model is observable;

[0118] Condition 5: The observer is asymptotically stable;

[0119] Condition 6: No constraints are active at steady state;

[0120] Condition 7: The optimization problem (7) has a feasible solution at all times.

[0121] Under the condition that Conditions 1 to 7 are satisfied, the unbiased model predictive control of the SCR denitration system based on dynamic variable gain Kalman filtering is analyzed:

[0122] Proposition 1: If the Kalman filter is asymptotically stable, the Kalman filter gain of the disturbance estimation is row full rank, that is, rank(K kal,d ) = n d , n d is the number of disturbance variables;

[0123] Proof: Consider the cases of |NI| > δ and |NI| ≤ δ separately

[0124] Case 1: |NI| ≤ δ, fal(NI, α, δ) = NI / δ 1-α

[0125] Denote K fal = 1 / δ 1-α , It is obtained that:

[0126]

[0127] Substitute formula (9) into formula (11), and it is obtained that:

[0128]

[0129] Since the filter is asymptotically stable and (1, 0) is not a pole, it is obtained that:

[0130]

[0131] Thus, it is obtained that is row full rank, K kal,d is row full rank, that is, rank(K kal,d ) = n d ;

[0132] Case 2: |NI| > δ

[0133] Due to symmetry, consider the case of NI > δ. At this time, fal(NI, α, δ) = NI α , and perform a Taylor expansion of fal(NI, α, δ) at NI = α:

[0134] fal(NI, α, δ) = NI α ≈ δ α + αδ α-1 (NI - δ) = αδ α-1 NI + (1 - α)δ α#(14)

[0135] Let \(K\) fal2 =\(\alpha\delta\) α-1 , \(m=(1 - \alpha)\delta\) α , we get:

[0136]

[0137] Comparing Equation (15) with Equation (11), we get:

[0138]

[0139] Similarly, since the filter is asymptotically stable and \((1,0)\) is not a pole, then \(rank(K\) kal,d ) = n d ;

[0140] Proposition 2: Select \(n\) d = \(n\) y , \(n\) y is the number of output variables. If the closed-loop system and the filter are asymptotically stable, then the following equation holds:

[0141]

[0142] where are the estimated values of the state variables, the input variables, the output variables, and the disturbance at steady state, respectively;

[0143] Proof: Since the closed-loop system and the filter are asymptotically stable, combining with Equation (8), we get:

[0144]

[0145] Since \(n\) d = \(n\) y , \(K\) kal,d is a square matrix; at the same time, combining with Proposition 1, we get:

[0146]

[0147] Substituting Equation (19) into Equation (8), we get:

[0148]

[0149] Rearranging Equation (19) and Equation (20), Proposition 2 is proved.

[0150] The proofs of the above two propositions ensure that fKF-MPC can achieve unbiased tracking of the setpoint.

[0151] S6: Set the controller parameters, including the sampling time \(T\) s, the prediction horizon P, the control horizon M, the weight coefficient matrices Q and R, and the upper and lower limits U of the control variable and its increment in the constraint conditions max , U min , ΔU max , ΔU min ;

[0152] On the premise of satisfying the constraint conditions in (7), the controller solves the optimization problem at each sampling moment and outputs the calculated control variable u to the SCR system represented by the transfer function model.

[0153] Based on the above scheme, in order to verify the effectiveness and superiority of the method of the present invention, comparative example verification is carried out in this embodiment, specifically as follows:

[0154] In this embodiment, the unbiased model predictive control method based on dynamic variable-gain Kalman filtering (fKF-MPC) proposed by the present invention is compared with PID, the existing state-space model predictive control algorithm, and MPC combined with a disturbance observer (DOB-MPC) in terms of effects. Figure 3 is the comparison diagram of the control effects of suppressing disturbances under different control methods, Figure 4 is the comparison diagram of the estimation effects of disturbances by different observers. The unmeasurable disturbance acting on the system input in the case can be an actually occurring unmeasurable input disturbance, such as the change in ammonia water concentration, the nonlinear influence of the valve, etc., or a disturbance equivalent to the input end of other random disturbances acting on the denitration reaction process.

[0155] The parameters of the MPC controller are set as: T s = 10s, P = 100, M = 10, Q = 15, R = I, Q k = R k = P k = I, the coefficient of the DOB filter T f = 120, the parameters of the PID controller K p = -5, K i = -0.2, the parameters of the fal function α = δ = 0.5, the upper and lower limits of the input variable u max = 50, u min = -50, the upper and lower limits of the input variable increment Δu max = 5, Δu min = -5. The transfer function of the SCR system is set as:

[0156]

[0157] Such as Figure 3 and Figure 4As shown, when the disturbance first appears, the unbiased model predictive control based on dynamic variable-gain Kalman filtering (fKF-MPC) proposed by the present invention dynamically amplifies the gain of the filter when the innovation is small, and thus can estimate the disturbance faster and more accurately. This enables the control action to be taken in a timely manner just after the disturbance starts to occur, significantly reducing the dynamic deviation between the system output and the reference, and enhancing the disturbance rejection ability.

Claims

1. An unbiased model predictive control method for SCR denitration system based on dynamic variable gain Kalman filter, characterized in that, It includes the following steps: S1: Establish an augmented state space model of the SCR system; S2: Based on the augmented state space model of the SCR system, establish a k-step forward prediction model; S3: Based on the k-step forward prediction model, construct a performance index optimization function; S4: Design a Kalman filter that dynamically adjusts the gain according to the innovation; S5: Apply the designed Kalman filter to model predictive control to achieve unbiased tracking of the set value at steady state through model predictive control based on dynamic variable-gain Kalman filtering; S6: For the SCR denitration system, use the transfer function to describe the relationship between the ammonia injection amount and the outlet NOx concentration, set the controller parameters, perform optimization solution for the performance index optimization function, and output the calculated control variable value to the transfer function model.

2. The unbiased model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 1, wherein, The expression of the augmented state space model of the SCR system in step S1 is as follows: where x k is the state variable of the system; u k is the input variable, i.e., the ammonia injection amount; y k is the output variable, i.e., the NOx concentration at the outlet of the SCR system; k represents the current moment; A, B, C are system-related matrices, and Ο is the zero matrix; Define the following variables: Then expression (1) is rewritten as:

3. The unbiased model predictive control method for the SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 2, characterized in that The establishment process of the k-step forward prediction model in step S2 includes: Define the increment of the input variable: Δu k = u k - u k-1 ; Considering the control time domain M and the prediction time domain P, and combining with Equation (3), the recurrence is obtained as follows: Where In the formula: is the predicted value of the output variable within the prediction horizon, and ΔU k is the increment of the input variable within the control horizon; M is the control horizon, P is the prediction horizon, and M ≤ P.

4. A bias-free model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 3, characterized in that, The performance index optimization function constructed in step S3 is as shown in formula (7): Among them, the performance index includes the deviation between the system output and the set value and the increment of the input variable; in the formula is the output of the prediction model, Q and R are weight coefficient matrices, and Y r is the set value of the output of the controlled object within the prediction time domain; ΔU max , ΔU min are the upper and lower limits of the increment of the input variable within the control time domain, respectively; U max , U min are the upper and lower limits of the input variable within the control time domain, respectively; I is the identity matrix.

5. A method for unbiased model predictive control of an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 4, characterized in that, The Kalman filter that dynamically adjusts the gain according to the innovation in step S4 is as shown in formula (8): wherein, is the estimated value of the augmented state variable, is the estimated value of the perturbation, and NI is the innovation of the Kalman filter; K kal,x is the gain matrix of the filter estimated state, and K kal,d is the gain matrix of the filter estimated perturbation.

6. A bias-free model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 5, characterized in that, Filter gain matrix K kal is calculated as shown in Equation (9): where P is the covariance matrix, Q kal and R kal are covariance matrices, respectively reflecting the covariance situations of process noise and measurement noise; the definition of the fal function is shown in formula (10): Where, α and δ are the parameters of the function, and e is the deviation; δ defines the size of the nonlinear interval, and α is used to adjust the nonlinear degree.

7. A bias-free model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 6, characterized in that The conditions for achieving unbiased tracking of the set value by the model predictive control based on dynamic variable-gain Kalman filtering in step S5 are as follows: Condition 1: The set value asymptotically converges to a constant value, the closed-loop system is asymptotically stable and converges to a constant value; Condition 2: The augmented system described by formula (1) can be stabilized; Condition 3: The number of disturbance variables is not greater than the number of system outputs; Condition 4: The augmented model can be observed; Condition 5: The observer is asymptotically stable; Condition 6: No constraints act at steady state; Condition 7: The optimization problem (7) has a feasible solution at all times.

8. A bias-free model predictive control method for an SCR denitration system based on dynamic variable-gain Kalman filtering according to claim 7, characterized in that, In step S5, analyze the unbiased model predictive control of the SCR denitration system based on dynamic variable-gain Kalman filtering: Proposition 1: If the Kalman filter is asymptotically stable, then the Kalman filter gain of the disturbance estimation is row full rank, i.e., rank(K kal,d ) = n d , where n d is the number of disturbance variables; Proof: Consider the cases of |NI|>δ and |NI|≤δ separately Case 1: |NI| ≤ δ, fal(NI, α, δ) = NI / δ 1-α Let K fal = 1 / δ 1-α , It is obtained that: Substitute formula (9) into formula (11) to obtain: Since the filter is asymptotically stable and (1,0) is not a pole, it is obtained that: It follows that is row full rank, K kal,d is row full rank, that is, rank(K kal,d ) = n d ; Case 2: |NI|>δ Due to symmetry, consider the case where NI > δ, in which case fal(NI, α, δ) = NI α , perform a Taylor expansion of fal(NI, α, δ) at NI = α: fal(NI,α,δ) = NI α ≈δ α +αδ α-1 (NI - δ) = αδ α-1 NI+(1 - α)δ α #(14) Let K fal2 = αδ α-1 , m = (1 - α)δ α , it follows that: Compare formula (15) and formula (11) to obtain: Similarly, since the filter is asymptotically stable and (1, 0) is not a pole, rank(K kal,d ) = n d ; Proposition 2: Select n d = n y , where n y is the number of output variables. If the closed-loop system and the filter are asymptotically stable, then the following equation holds: wherein, are respectively the estimated value of the state variable, the input variable, the output variable, and the disturbance at the steady state; Proof: Since the closed-loop system and the filter are asymptotically stable, combined with formula (8), it is obtained that: Since n d = n y , K kal,d is a square matrix; at the same time, combining Proposition 1, it can be obtained that: Substitute formula (19) into formula (8) to obtain: Arrange formula (19) and formula (20), and Proposition 2 is proved.

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